The electromagnetic wave evolution on very long distance

Transcription

1 The eletromagneti wave evolution on very long distane Pierre Réal Gosselin Abstrat We explain the radiation s redshift Z from far away Galaxies using only Maxwell s lassial equations and the energy onservation priniple. Hubble s law sprouts out naturally as the onsequene of the transformation of this radiation on long distanes. We ompute the onstant H 0 (84,3 Km s 1 Mp 1 ) from the Pioneer satellite data and explain its anomalous behaviour. We resolve some situations that are still enigmati for the atual osmology. We review the distane modulus formula and evaluate the limits of osmologial observations. 1

2 Contents Contents 2 1 Introdution 3 2 Theory Extreme propagation I Extreme propagation II Wavelength evolution The Hubble onstant Pioneer H Galaxies Solved enigmas Reeding speed of the Cepheids Luminosity inrease Cosmi mirowave bakground and supernova Distane Distane modulus The world we an see Conlusion 22 Referenes 23 2

3 1 Introdution I nterpreting the redshift Z of the radiation oming from the distant galaxies as a Doppler effet implies that those galaxies are moving away from the observer. In 1929 Edwin Hubble [1] [2] showed that the reeding speed v of the observed galaxies was proportional to their distane z from the observer, was isotropi and related by the proportionality onstant H 0. In the following years, this onstant has been evaluated to 73 kilometres per seond per Mega parse but reently revised aording to Plank satellite s data [3] to 67 kilometres per seond per Mega parse. Hubble law is written as, v = zh 0 (1.1) where v is the soure reeding speed, z its distane from the observer and H 0 the proportionality onstant. The redshift Z is a measure of this speed relative to light s vauum speed Z = v (1.2) so that the distane is given as z = Z H 0 (1.3) Relative to the soure wavelength λ 0 and the observed wavelength λ, the redshift is Z = λ λ 0 λ 0 (1.4) Suh interpretation infers an expanding universe sine all observable objets seems to speed away the farther they are from the observer. Conversely, this implies [4] that 13,7 billions years ago ( 1 H 0 ), all the universe was embedded in a singularity that exploded to produe the expanding universe that we are observing today. T he Doppler effet being the ratio of soure s speed to light s speed and the fat that nothing an exeed the speed of light this ratio must always be lower than one. But it is ommon to observe galaxies showing Z ratios greater than one up to values of 12 aording to the more reent observations [5] [6]. This goes against interpreting the redshift as Doppler aused and also against an expanding universe. B ut sine Hubble disovery [1] and oupled to Lemaître thesis [7] [8] [9] [10], nearly all osmology theoretiians agree on a new kind of universe expansion. This is no more an explosion of matter into spae but the more esoteri onept of a spae expansion whih is seen through the general relativity glasses. The expanding spae idea explains the redshift by the strething the light rays suffers during their travel through spae. Considering the elastiity of spae is mere speulation beause there aren t any experienes possible to prove it. This is an open door to all kinds of exoti universe models and even to questioning the known observed properties of matter : Cameron [11], Terazawa [12]. 3

4 O bserving light or photons in our loal universe as well as in the laboratory shows us that photons are partiles or waves that keep their properties indefinitely. On the ontrary, atomi partiles have a measurable lifetime and an deay into other partiles. Sine the light speed is the maximum speed of any interation that may happen in the universe this implies that the photon annot suffer any other ation exept to move. Then time doesn t exist for the photon and it is immutable. N othing may suggest that physi s laws are different at large distane from us than in our loal environment. If we agree on the fat that there aren t any differene between the loal universe and the most remote one and also between, then it is advisable to onsider that the photons might suffer a kind of transformation between the emission point and the observer. This way the redshift an be justified differently than by the strething of spae. The sole laboratory that an permit this verifiation is the universe itself sine the billion of years required. We propose to explain the redshift and Hubble law through suh a slow transformation of the photons en route. Aording with the afore mentioned immutability of the photon, it must be must oneded that the maximum speed of any interation in the universe is a little bit higher than the photon speed. This limit might be very lose to the light speed beause of the extremely long time required for photon transformation and the fat that all experiments done up to now are very well explained using the speed of light as the maximum speed limit of interations in the universe. Then the photon is subjet to strutural transformation like any other denizen of the universe. This proposal seems to us muh more aeptable and less esoteri than the elastiity of the spae O n an other side, onsider the eletromagneti radiation that omes from all parts of the sky. It has a wider spetrum muh larger than the optial one. Partiularly, radio astronomers A. Penzias et R. Wilson [13] in 1964 disovered a uniform and isotropi radiation at a mirowave frequeny of 160,2 GHz or 1,873 mm in wavelength orresponding to a temperature of 2,72548 K [14]. This radiation ouldn t be assoiated with any objet of the sky and it s presene has been explained as a residual of the Big Bang that happened 13,7 billions years ago. Very energeti photons emitted at that time might have lost their energy through spae expansion and the strething of their wavelength. Today we would observe them as a osmi mirowave bakground residual (CMB). W e already have proposed the transformation of the photons en route but this proess mustn t proeed indefinitely, it must end somewhere. If it didn t, it would end up with an infinite number of photons of zero energy, an unaeptable situation in nature. We then propose that this endpoint of transformation happens when the photon energy reahes the CMB level. This way all radiation oming from everywhere melts in a kind of uniform fog that makes the physial limit of the observable world. 4

5 2 Theory A ording to our hypothesis, we proeed from the priniple of energy onservation and built following two methods. The first one makes use of Maxwell s eletromagneti field equations while the seond one is based on a sequene of suessive photon mutations. Both gives the same results. Thereafter we onsider the evolution of the eletromagneti wave. 2.1 Extreme propagation I T he vauum properties of an eletromagneti wave at very far distanes are unknown to us. We suppose they are the same as they are loally meaning that Maxwell s laws of eletromagnetism are the same everywhere in the universe. Then for a plane wave moving in the diretion k, the eletrial field E and the magneti field H are dependent on distane z and time t. E = i E x (z,t) (2.1) E x = E exp [ jω(t z ) + θ] (2.2) H = j H y (z,t) (2.3) H y = H exp [ jω(t z ) + θ] (2.4) The Poynting vetor represents the energy flux arried by the wave whih is for the plane wave S = E H (2.5) S = E2 µ 0 k (2.6) Between extremely distant points the Poynting vetor annot represent the energy onservation priniple. A redshift is observed meaning a variation of the wavelength, an absent parameter of S. Meanwhile the Poynting vetor ertainly represents the mean energy arried by the photons of the wave eah being of energy E = ω (2.7) When onsidering extremely long distanes it is appropriate to onsider the variation of the photon density N and it s energy level so that the energy regime is preserved and the quantity ξ = N ω (2.8) 5

6 is kept onstant on long distanes while N and ω vary aording to the distane z. Bringing together those two quantities S = ξ (2.9) E 2 = N(z)ω(z) (2.10) µ 0 E = (µ 0 N(z)ω(z)) 1 2 (2.11) shows an eletrial field E varying as a funtion of the photon density N and the irular frequeny ω both being dependent on the distane z. Then the omponents of the eletromagneti wave are Simplifying the writing and knowing that we get Sine that at any point in spae then E x = (µ 0 N(z)ω(z)) 1 2 exp[ jω(z)(t z ) + θ] (2.12) H y = ( N(z)ω(z) ) 2 1 exp[ jω(z)(t z ) + θ] µ 0 (2.13) E x = F z exp [ jω z (t z ) + θ] = F zexp [ ] (2.14) H y = G z exp [ jω z (t z ) + θ] = G zexp [ ] (2.15) E x z = µ H y 0 t E x z = F z z exp [ ] + jf z exp [ ] { ω z z (t z ) ω z } H y = G z exp [ ] { } jω z t (2.16) (2.17) (2.18) E H = F z G z = µ 0 (2.19) G z = F z µ 0 (2.20) Considering F z z + jf ω z z z (t z ) = 0 (2.21) E = F = (µ 0 N z ω z ) 1 2 (2.22) 6

7 one get { N z ω z z + ω z } { N z ω z + j 2N z ω z z z (t z } ) = 0 (2.23) The real part between braes may be obtained differently by onsidering the fat that the quantity ξ (2.8) doesn t vary with distane so it s derivative is null and we get The solution of this differential equation is where giving ξ z = N ω z z z + ω N z z z = 0 (2.24) N z = α e z η +C 1 (2.25) ω z = β e z η +C 2 (2.26) N z z = α η e η z (2.27) ω z z = β z η e η (2.28) α C 2 e z η = β C 1 e z η (2.29) This last equation being true for any value of z implies C 1 = C 2 = 0. The limiting onditions are at z = 0 : N z = N 0, ω z = ω 0 so that and finally where the wavelength is The redshift is and for very short distanes z η α = N 0 (2.30) β = ω 0 (2.31) N z = N 0 e z η (2.32) ω z = ω 0 e z η (2.33) λ z = λ 0 e z η (2.34) Z z = e z η 1 (2.35) Z z = z η (2.36) 7

8 Here we have Hubble s law whih in order to be reognized under it s lassial form we hose and generally we have for the distane saling of the redshift η = H 0 (2.37) Z z = e zh 0 1 (2.38) Inversely the distane expressed as a funtion of the redshift beome and for the wavelength and the photon density z = H 0 ln(z + 1) (2.39) λ z = λ 0 e zh 0 (2.40) N z = N 0 e zh 0 = N 0 (Z + 1) (2.41) This shows that the alulation of osmi distanes using the lassial Hubble law leads to overestimate real distanes and that a logarithmi sale is the due way. The soure wavelength and intensity grows linearly aording to the redshift or exponentially for the distane. Figure 1 shows the relationship between the intensity and the wavelength as a funtion of the redshift for a Gaussian mimiking a spetral line. Any spetrum keeps its struture while its wavelength (2.40) and intensity (2.41) grows as a funtion of distane. The apparent reeding speed is exponential as per equations (2.39) and (1.2). z = ln( v + 1) (2.42) H 0 { {zh 0 v = e } } 1 (2.43) 2.2 Extreme propagation II S petral properties of atoms are well known in the laboratory. But when we observe them from far distanes they show a redshift of their wavelength. Individual photons are haraterized by a wavelength λ 0 and energy E 0 E 0 = h λ 0 (2.44) Let us onsider a ohort of N 0 photons per unit volume showing an energy per unit of volume G 0 G0 = N0E0 (2.45) 8

9 Figure 1: Spetral line evolution After a time T there are N k > N 0 photons per unit volume showing an energy per unit of volume G k Aording to the priniple of onservation of energy we write G k = N k E k (2.46) N k E k = N 0 E 0 (2.47) Here we suppose that the transformation of the photons happens by suessive leaps. A first one produes a new photon and the energy in the group is re-equilibrated. This proess is the same for all other transformations happening in the group. After k transformations the energy of N 0 + k photons beomes E k = N 0E 0 N 0 + k After k transformations the number of new photons is { } E0 k = N 0 1 E k { h } λ k = N h λ k (2.48) (2.49) (2.50) λ k λ 0 k = N 0 λ 0 (2.51) k = N 0 Z k (2.52) 9

10 where we made use of the redshift Z k. The photon density is then N k = N 0 + k (2.53) N k = N 0 + N 0 Z k (2.54) N k = N 0 (Z k + 1) (2.55) The spetral line intensity I k is proportional to the number of photons per unit of volume and it also inreases the same way as and the photon energy is I k = I 0 (Z k + 1) (2.56) and the wavelength is E k = E 0 Z k + 1 (2.57) λ k = λ 0 (Z k + 1) (2.58) B etween suessive transformations the ohort moves a distane z. Without saying anything about the way suh transformations happens, we suppose that the tension pushing the photons to transform is proportional to the atual photon density N k. The number of new photons k per unit of distane is and inversely, the distane of transformation is given by k z N k (2.59) z k 1 N k (2.60) whih says that the distanes where the transformations happens are inversely proportional to the photon density whih onstantly inreases upon distane. If b is the proportionality onstant and N k = N 0 + k we have for small intervals Upon integration z k = b N 0 + k The initial onditions being z = 0 and k = 0 then (2.61) z = b ln(n 0 + k) +Cte (2.62) Cte = b ln N 0 (2.63) and the distane is z = b ln N 0 + k N 0 (2.64) 10

11 Using equation (2.52) we write z = b ln (Z + 1) (2.65) from whih the redshift and the wavelength as a funtion of distane are Z = e z b 1 (2.66) λ z = λ 0 e z b (2.67) Expanding the exponential as a series and keeping the first order terms for small distanes Z = z b (2.68) There we reognize Hubble s law and we set b = H 0 (2.69) From this point we ompute the same equations as in the preeding setion that is to say equations (2.38), (2.39), (2.40) and (2.41). 2.3 Wavelength evolution T he wavelength transforms as an exponential of the distane (2.40). Aording to our model, the wavelength onverge through the osmi radiation bakground wavelength λ mb showing a redshift Z mb = λ mb λ 0 λ 0 (2.70) Up to this point we onsidered an energy derease of the photons. We make the hypothesis that for photons less energeti than the CMB, there is an energy inrease while their number density dereases. The wavelength of those photons dereases until it opes with the CMB one λ mb and a blue shift shall be observed for those. Consequently it shall be better to speak of a osmi-shift that will be > 0 for red-shift or < 0 for blue-shift. T he evolution of any wavelength aording to distane is given by the following expression λ(z) = λ mb (λ mb λ 0 ) e H 0 z (2.71) and the osmi-shift is Z(z) = Z mb (1 e H 0 z ) (2.72) Figure 2 shows the wavelength evolution through the mb. 11

12 3 The Hubble onstant Figure 2: Wavelength evolution T he enigmati deeleration of the Pioneer satellite onfirms our model of the spatial transformation of the eletromagneti wave. It gives us the opportunity to measure diretly the value of the Hubble onstant. The proedure might also be applied to galaxies and stars. 3.1 Pioneer T he Pioneer 10 satellite has been deelerating onstantly sine it s departure from the solar system and still was when ommuniations ended due to the loss of strength of the signal, Turyshev and Toth [15]. The Doppler signal measuring the satellite speed drifted onstantly showing a deeleration of the satellite. Sine the satellite was out of solar bounds it should have kept a onstant speed and up to now no satisfatory explanation has been given to this phenomena. T he satellite distane and speed were measured very preisely by observing a S band signal of frequeny 2,1 GHz sent from earth station and returned as 2,3 GHz by the satellite in suh a way that the stability and preision of the signal were independent of the satellite equipment. The satellite being out of solar bounds should have moved ballistially aording to the lassi mehanial laws. Throughout the whole journey, a onstant frequeny drift of 5,99 ± 0, Hz se 1 has been observed toward a higher one. Interpreted as a Doppler shift, it is equivalent to a satellite deeleration of 8,74 ± 1, m s 2. We onsider that this signal variation is nothing else than the effet of the transformation of the eletromagneti signal aording to our model. 12

13 C learly if the satellite slows down, one will observe a blue shift of the Doppler signal whih is already red shifted beause of the satellite reeding speed. Newtonian mehani tell us that the satellite doesn t slow down but moves at onstant speed. The signal round trip is inreasing at a onstant pae and aording to our model, the signal must suffer a onstant hange. Sine the mean frequeny of the signal at 2,2 GHz is lower than the osmi mirowave bakground of 160,2 GHz, the frequeny of the signal must inrease or equivalently the wavelength shorten. So the observed blue shift drift owing to the ontinuous inreasing signal round trip distane. And the false impression of a slowing down of the satellite. 3.2 H 0 T his signal shift permits us to ompute diretly the Hubble onstant. Let us take the time derivative of equation (2.71) and taking into aount equation (2.40) and the following z = t and λ = /ν λ = H 0 (λ mb λ 0 ) e H 0t λ = H 0 Z mb λ 0 e H 0t (3.1) (3.2) λ = H 0 Z mb λ (3.3) H 0 = λ λ 1 (3.4) Z mb H 0 = ν ν 1 Z mb (3.5) Referring to the Pioneer satellite data, we selet as the mean frequeny the value of 2,22345 GHz whih is between 2,1 GHz, and 2,3 GHz. Using (2.70), the osmi shift is Z mb = ν 0 ν mb ν mb (3.6) 2, ,2 Z mb = = 0, ,2 (3.7) In this ase the reeption frequeny is nearly the same as the emission one so the Hubble onstant value is H 0 = Using 1 Mp = 3, Km 5, , , = 2, se 1 (3.8) H 0 = 2, , = 84,29852 Km se 1 Mp 1 (3.9) H 0 = 84,3 Km se 1 Mp 1 (3.10) 13

14 3.3 Galaxies I n the ontext of a slowing down expanding universe, Loeb [16] has proposed to measure the variation of the redshift of galaxies as a funtion of time. Sine there isn t any suh phenomenon but only a distane hange due to the peuliar veloity of the galaxies in the diretion of observation, there will be a measurable red-shift or blue-shift independently of the distane of those galaxies. Long term observations depending on the intrinsi speed of the galaxies will be needed. I f we an follow the frequeny drift of a signal from a moving satellite and then derive the value of the Hubble onstant, we an onsider it possible for galaxies. Then for any galaxy at any distane, monitoring over many years the the drifting of a spetral line suh as Lyman α or H α will enable us to ompute Hubble onstant using equation (3.4). Also it will be easy to hek the onstany of the Hubble onstant as a funtion of distane. Using equations (3.4), (2.70), (1.4), the intrinsi speed of the soure v int and the extra time taken by the soure T and by light t λ = λ H 0 Z mb (3.11) λ = t λ H 0 Z mb (3.12) t = 2 v int T (3.13) λ = 2 T v int H 0 Z mb λ 0 (Z os + 1) (3.14) If during T = 10 years, we observe the spetral line λ 0 = H α = 6563 from a galaxy situated at a redshift of Z os = 1 for whih it is estimated it has a reeeding speed of v int = 1000 km/s and using for the Hubble onstant the value of H 0 = 84,3 km/se/mp, a drift of about 2, shall be observed. The same proedure might be applied to stars having a better visibility. 14

15 4 Solved enigmas M ore and more deviations or unexplained effets pop up in the ontext of an expansionist osmology. Some of those phenomenons are very well explained by our model and here three are analysed. 4.1 Reeding speed of the Cepheids W e have shown that the apparent reession speed is exponential and not linear (2.43). If a linear relation is kept (1.1) when observing objets situated at farther and farther distanes or inreasing red-shifts, higher and higher values of the Hubble onstant H 0 will be found. This explains the differene between Cepheids lose to us and others farther from us. This fat is shown and disussed in the paper of Arp [17] where he looks for an explanation by an exess of redshift for the distant Cepheids. Figure 3 reprodues figure 4 of his paper where the inreasing values of the Hubble onstant as a funtion of distane are learly seen. 4.2 Luminosity inrease I n a study of two ultra and hyper luminous galaxy groups (Lyman Break galaxies) showing high redshift, Oteo and al [18] find that all galaxies of a group at Z 1 have a magnitude less than 11,7 and all those of an another group at Z 3 have a magnitude greater than 12,4 while both groups were onstruted to make two homogeneous populations with idential properties. Those investigators questions the possible influene of the redshift on the evolution of the far infrared radiation FIR oming from those galaxies. Considering our model, it is lear that the observed luminosity inreases as a funtion of it s redshift (2.41). In this ase, the redshift ratios between the two groups is simple to double (3 + 1)/(1 + 1) = 2. The luminosity will show the same ratio or in term of magnitude it will translate to a differene of 2,5 log(2) = 0,753. This is the observed magnitude differene between the two groups [> 12, 4] [< 11, 7] = [> 0, 7]. 4.3 Cosmi mirowave bakground and supernova Y ershov and al. [19] has showed a high orrelation between the loal inrease of the osmi mirowave bakground temperature T sn at supernova positions and the redshift of those supernova Z sn. Looking at SN type Ia they find that the temperature inreases as T sn = 58,0 ± 9,0 Z sn [µk]. This loal temperature exess is proportional to the assoiated redshift of those supernova. The expansionist osmology annot explain this phenomenon. However this effet onfirms our transformation model of the eletromagneti energy as a funtion of distane. At those supernova spots, there is always an exess of temperature over the osmi bakground. And this inrease is diretly proportional to the soure s distane or its red-shift. S upernova are onsidered osmi standard beause they all have the same properties and produe idential energy spetra, Coelho [20]. Let us onsider the energy spetra of suh supernova where we define a small wavelength band λ 1 λ 2 braing the CMB 15

16 Figure 3: Hubble onstants as a funtion of distane for Cepheids 16

17 wavelength λ mb. We suppose that the photons density I(λ) into that interval is onstant I 0 with the photon energy as h λ. The total energy in that band is E = λ2 λ 1 I 0 h λ dλ (4.1) E = I 0 h { ln(λ 2 ) ln(λ 1 ) } (4.2) At a distane d or a red-shift Z, we observe this band slightly ontrated around the CMB wavelength λ mb that is between λ 1 and λ 2. Using (2.71) and (2.40) we get λ 1 = λ mb (λ mb λ 1 ) e H 0 d (4.3) λ 1 = λ mb (λ mb λ 1) e H 0 d (4.4) λ 1 = λ mb {(1 λ mb λ } 1 (Z + 1) (4.5) λ mb { ln(λ 1 ) = ln(λ mb ) + ln 1 λ mb λ } 1 (Z + 1) (4.6) λ mb { ln(λ 2 ) = ln(λ mb ) + ln 1 λ mb λ } 2 (Z + 1) (4.7) λ mb { ln(λ 2 ) ln(λ 1 ) = ln 1 λ mb λ } { 2 (Z + 1) ln 1 λ mb λ } 1 (Z + 1) (4.8) λ mb λ mb and making an approximation of the logarithm by its argument A substitution in (4.2) gives ln(λ 2 ) ln(λ 1 ) = (Z + 1) λ mb (λ 2 λ 1) (4.9) E = I 0 h (Z + 1) λ mb (λ 2 λ 1) (4.10) The instrument measure a temperature inrease T onverted through the Boltzman onstant as an energy inrease E = k B T (4.11) Then we get for the temperature inrease T = I 0 h (λ 2 λ 1 ) k B λ mb (Z + 1) (4.12) Our model predits a temperature inrease proportional to the red-shift and is orroborated by the Yershov paper. This paper points out that the measure made at the frequeny of 143 Ghz or 1,67 mm shows the best orrelation for a oeffiient value of 67,6 ± 6,3 µk. If we suppose a band of 0,8 mm and use that wavelength as a substitute for the CMB wavelength we get T = 1, , , , I 0 (Z + 1) (4.13) T = 1093 I 0 (Z + 1) µk (4.14) 17

18 5 Distane F ollowing the eletromagneti wave distane transformation, we need to review the distane modulus. We evaluate the maximum dimension of the observable world. 5.1 Distane modulus L et us onsider the inreasing photon density (2.41) N z = N 0 e zh 0 (5.1) A monohromati soure of luminosity L 0,λ0 at a wavelength λ 0 will look at a distane z as a longer wavelength λ z (2.40) λ z = λ 0 e zh 0 (5.2) Suh soure will produe a flux S z,λz that an observer will measure as proportional to the inrease of the photon density and inversely proportional to the squared distane S z,λz = L 0,λ0 e zh0 (4πz 2 ) (5.3) At two different distanes d et f the orresponding fluxes will be S d,λd and S f,λ f whose ratio is S d,λd = L 0,λ0 S f,λ f = L 0,λ0 e dh0 (4πd 2 ) e f H0 (4π f 2 ) (5.4) (5.5) ( ) S d,λd f 2 = e H 0 (d f ) (5.6) S f,λ f d For this soure, the magnitude differene between those two points aording to the definition of magnitude is m d m f = 2,5 log S d,λ d (5.7) S f,λ f {( ) f 2 } m d m f = 2,5 log e H 0 (d f ) (5.8) d At the distane f = 10 p, m f beome the onventional referene value for the absolute magnitude M. This magnitude differene is the definition of the distane modulus µ and then we have for this soure µ = m M = 5 log d p 5 1,086 H 0 18 {d p 10 p } (5.9)

19 Using equation (2.39) { } µ = 5 log ln (Z d + 1) 5 1,086 H { } 0 ln (Z d + 1) 10 p H 0 H 0 (5.10) Negleting the very small value of 10 p and using H 0 = 84,3 Kmse 1 Mp 1 and H 0 = 3,5563 Gp we obtain the expression for the distane modulus different from the lassial one µ = 42, log ln (Z d + 1) 1,086 ln (Z d + 1) (5.11) µ = 42, log Z d (5.12) Table 1 shows the distane d and the distane modulus µ against the lassial values as a funtion of the redshift Z. The distane modulus grows up to a maximum at Z = 6,38 and then dereases slowly. Inluded are the orresponding values of the expansionist model obtained fron Nik Gnedin alulator [20] using H 0 = 67,3 and Ω 0 = 0,315. Figure 4 illustrates this table. Figure 4: The distane modulus (upper urves), the distane ( Gp, lower urves) the wavelength (H α 10 5 m, exponential) and the lassial values (dotted urves). 5.2 The world we an see O ur model shows photon transformation along distane, ending when the photon energy orrespond to the CMB radiation. At this point photons have a wavelength 19

20 Redshift Distane Distane Classial Classial Distane Module modulus distane modulus expansion Nik Gnedin model alulator (2.39) (5.11) (1.1) (5.12) Z d d Gp µ d Gp µ d Gp µ 0,1 0,34 37,54 0,36 37,75 0,48 38,42 0,2 0,65 38,86 0,71 39,26 1,02 39,85 0,3 0,93 39,56 1,07 40,14 1,61 41,04 0,4 1,20 40,02 1,42 40,76 2,25 41,77 0,5 1,44 40,35 1,78 41,25 2,94 42,34 1 2,47 41,20 3,56 42,75 6,81 44,17 2 3,91 41,76 7,11 44,26 15,97 46,02 3 4,93 41,96 10,67 45,14 26,07 47,08 4 5,72 42,04 14,23 45,76 36,72 47,82 5 6,37 42,07 17,78 46,25 47,75 48,39 6 6,92 42,08 21,34 46,64 59,06 48,86 6,38 7,11 42,09 22,69 46,78 63,42 49,01 7 7,40 42,08 24,89 46,98 70,60 49,24 8 7,81 42,08 28,45 47,27 82,31 49,58 9 8,19 42,06 32,01 47,52 94,16 49, ,53 42,05 35,56 47,75 106,14 50, ,83 41,86 71,13 49,26 230,36 51, ,21 41,70 106,69 50,14 359,09 52, ,21 41,57 142,25 50,76 490,15 53, ,98 41,46 177,82 51,25 622,76 53, ,41 41,06 355,63 52, ,28 55, ,86 40,62 711,26 54, ,14 57, ,30 40, ,89 55, ,21 58, ,32 40, ,53 55, ,00 58, ,11 39, ,16 56, ,02 59, ,57 39, ,31 57, ,98 60, ,03 38, ,63 59, ,69 62, ,47 38, ,94 60, ,76 63,14 Table 1: Distane and distane modulus as a funtion of redshift aording to our model, the lassial model and the expansionist model. 20

21 of 1, Å orresponding to a temperature of 2,72548 K. Considering the Hydrogen line H α = λ 0 = 6563 Å, photons at end of ourse will have a redshift of Z mb = λ mb λ 0 = 1, = 2853 (5.13) λ The orresponding transformation distane z = d aording to the modified Hubble law (2.39) where H 0 = 84,3 Km s 1 Mp 1, = Km s 1, 1Mp = 3, Km and 1p = 3,26 al is d mb = H 0 ln {Z mb + 1} (5.14) d mb = ln { } Mp 84,3 (5.15) d mb = 28,3 Gp = 92,3 Gal (5.16) The CMB represents the true limit of knowledgeable universe, the maximum dimension of the observable universe not its physial dimension. This distane vary upon the wavelength of the photons. It is around 92,3 Giga light years if we onsider the H α hydrogen line and 194,3 Giga light years if we onsider gamma rays. Table 2 shows some values very different from the usual lassi value of 13,7 Giga light-years whih is nearly thirteen times smaller than the knowledgeable universe. Line λ 0 [Å] Z mb d mb [Gp] d mb [Gal] L α ,3 111,9 L ,3 115,2 H α ,3 92,3 H ,4 99,1 γ 1 1, ,6 194,3 Table 2: Transformation distanes L et us look at Quasars whih are of a very great luminosity and are usually very far away objets. A value of Z = 3,638 has been measured for Quasar Q that put it at a relative distane of d ln(1 + 3,638) = = 0,1928 (5.17) D ln( ) It is about 1/5 the theoretial observable limit or 5,46 Gp (17,8 Gly). ULAS J shows a Z = 7,1 and is relatively plaed at 26% that is 7,4 Gp or 22,9 Gal 21

22 6 Conlusion T he expansionist model of osmology also alled the "Big Bang" is a speulative one. Instead of ompounding with an elasti relativisti metri with adjustable parameters, we find more plausible our model based exlusively on Maxwell eletromagnetism and the quantum world. Contrarily to tired light models it doesn t blur images but enhanes their luminosity while reddening them. O ur model shows that osmologial distanes an be measured aording to a logarithmi law of redshift. It gives a sound basis to the Hubble onstant whih we evaluate to 84,3 Km se 1 Mp 1 diretly from the Pioneer satellite data. And at the same time it solves the enigma it posed. W e reviewed some problemati ases for the expansionist model and showed that they are naturally explained by our model. W e reviewed the distane modulus aording to our model and set new frontiers to the knowledgeable universe. The world is not physially limited to 13, 7 billion light-years but knowledgeable up to 100 billion light-years. 22

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